For a testing method, there is **sensitivity** <true positive rate = P(tested positive given that the situation is true)> and **specificity** <true negative rate = P(tested negative given that the situation is not true)>.
My question is why **specificity** is not defined by something like P(the situation is true given a positive test result), doesn’t that also tell us whether untargeted situations trigger positive?
In: Biology
Misread the OP
If you had a lopsided sample, with lots of “true” observations, your measure does not look bad when it is supposed to look bad if you guessed all observations are true.
Let’s think of it another way: Imagine you are sick with a respiratory illness. You suspect you may have COVID, so you take an antigen test for COVID.
Let’s imagine first that you do in fact have COVID. *Sensitivity* is the test accurately telling you so. If it tells you that you don’t have COVID when you in fact do, that’s a False Negative (Type II Error). It’s literally telling you how good the test is at detecting the thing.
On the other hand, let’s imagine you don’t have COVID, and instead have something else, like Norovirus. *Specificity* is the test accurately ruling out COVID in the case where it doesn’t apply. If it tells you that you do have COVID when you in fact don’t, that’s a False Positive (Type I Error). It’s literally telling you how good the test is at only responding to the thing and not being triggered by something else.
In both cases, sensitivity and specificity are defined by the probability of the test being correct given the underlying reality. False Negative/Positive are the inverse of those states.
Think about the denominator for specificity. It includes all of the people who don’t have the disease/condition. This means it is the true negative *and* the false positives. If a test is not very specific, that means there will be a lot of false positives, i.e. that people are getting positive results for not having the disease, or having a different disease.
The P(disease given positive test result) is the positive predictive value. This changes depending on how many people have the disease in the first place. If the disease is really rare, then most positive results are false positives. If the disease is really common, then most positive results are true positives. The sensitivity and specificity don’t depend on how common the disease is or not
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